Lesson 2.10

Factoring by Grouping

When a polynomial has four terms, standard factoring techniques fail. Grouping lets you split the polynomial in half, factor each half, then extract a common binomial factor.

Introduction

You already know how to factor out a GCF and how to factor trinomials. But what about polynomials with four terms? There's no trinomial pattern to apply. Enter: factoring by grouping.

Past Knowledge

You can factor out the GCF from any polynomial expression.

Today's Goal

Factor 4-term polynomials by splitting them into two groups and extracting a common binomial factor.

Future Success

Grouping is essential when factoring polynomials generated by the Rational Root Theorem later in this unit.

Key Concepts

1. The Strategy

Given a 4-term polynomial, split it into two groups of two and factor the GCF from each group. If the leftover binomials match, you're done!

2. The Steps

1
Group the polynomial into two pairs of terms
2
Factor out the GCF from each group
3
Check: do the remaining binomials match?
4
Factor out the common binomial

💡 What If They Don't Match?

Try rearranging terms or factoring out a negative from one group. If it still doesn't work, the polynomial may not factor by grouping.

Worked Examples

Example 1: Standard Grouping

Basic

Factor .

1

Group

2

Factor Each Group

3

Factor the Common Binomial

Both groups contain . Factor it out:

Example 2: Factoring Out a Negative

Intermediate

Factor .

1

Group

2

Factor Each Group

Factor from the second group to make the binomials match.

3

Factor the Common Binomial

Example 3: Rearranging First

Advanced

Factor .

!

Rearrange

The terms as-is don't group nicely. Rearrange to pair terms that share a GCF:

1

Group & Factor

2

Factor the Common Binomial

Common Pitfalls

Forgetting to Factor Out a Negative

If the first attempt gives you and , they don't match! Try factoring from the second group to flip the sign.

Wrong Grouping

The first/second pair isn't always the right grouping. If the binomials don't match, try rearranging the terms and grouping differently.

Real-Life Applications

In engineering design, optimization problems often produce higher-degree polynomial equations. Factoring by grouping helps engineers decompose these into simpler factors — for instance, finding critical stress points in a beam modeled by a cubic polynomial.

Practice Quiz

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